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A Geometrical Vector Algebra
T. FROCTOK HALL. V ANCOUVFH. CANADA
9.
10.
U.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Introduction
Notation
Cosine A B
Addition and Sulttraction
Collinear Vectors... ,
Coplanar Vectors.
Multiplication
Collinear Multiplier ,
Perpendicular Multiplier
The Product A B
Permutation of Factors
Operand Di.stributive
Operator not Distributive
Factors not Permu** .ile
Powers of Operator
Laws of Multiplication
Perpendicular Factois
Division
A"B; General Formula
Quaternions
Vector Arcs
Sumof Circular and Straight
Vectors
Sum of Circular Vectors
Spherical Tiiangle
CONTENTS
fjlKi-
Vm-
... 1
2r).
('(inic W'ftfus
13
1
■m.
DiffervntiHtidn
14
... 2
27.
Ditrfi«*ntittl ("otffk-icnt of
■i
A"B
1.^1
. :l
2».
Curvature
1()
:i
29.
Linear Loci
\e,
;t
:)o.
Surface Loci
11.
... 4
m.
Solid Loci
17
4
32.
rommon Rejjions
17
4
38.
Projections
20
5
34.
Plane Algebra
21
5
31).
Four-?pace Algebra
21
6
:«.
Multifilication in 4-space
21
6
37.
Perpendicular Vectors in
,, , 7
4-space
22
7
38.
Coplanar Equations
23
7
39.
Perpendicular to a Vector
23
S
40.
Normal to a Plane
23
8
41.
Normal to a 3-flat
24
9
42.
W. V. F and N
25
10
43.
The Product Ab"c .
25
tht
44.
Quaternion Rotors
20
, 10
45.
Intersecting Loci
26
11
4«.
Projections
27
12
47.
Projections of a Regular
Tessaract on a :i-Hat
or AVB^
A (Seontrtriral Drrtur Algrbra
By T PKOCTOK HAIL. MA., I'll I) . Ml).
I. Th» lawi of oppratiun of any alitihra :irt' ulllmi''.|.v haHocI upon Iti
deflnltions. If tht> di>flnitionM arc Kt'omctrji-al theal;,(-l>rj .-o[ 'i-aviontt have
Kt^omvtric curroiwnilt'no-ii. Thf opei .lions of ailililion and .tuhtractlon in
common algebra, for vxampli*. corrfH[io ;o the ^t'omt'tiic ail(li:ion and
subtraction of atraJRht lines, vectora. aurfacc.*, etc.
In thin alffehra nt>w dcnnitionR of vector niultiplication and diviHion arc
adopted, in '-onaequence of which all alKehraic iiiaTuMonK ujion vectora
(directed uniocated atrainht linea or Btepa), or rather upi.n victor symMa.
correspond to xeometric operationa ir apace Ujain the vectors themaelvea;
IU1.I every alirebraic vector expreatiim eoire^pcinila l<> aome Keometric
confi^ration of the vectora themaelvea.
In every vector demonatratlon or problem, therefore, the atudtnt may
think in terms of either alKebra tr geem.'try or btith ; and may at any time
change from one realm of thot to the other with no break in the continjity.
This algebra is developed first in terms of analy'Lal geometry for
threefold apace, and is then adapted to two-fold and to four-fold apace.
Complex numbers, spheiical trigonometry, and quaternion rotations, apia-ar
as special cases.
2, NOTATION.— Taking three rectangular axes X, Y, Z, let j», y, >
denote unit vectors (steps) outward from the centre (), along the axes.
|Unit vectora in the opposite direction fr-om O are denoted by jr. y. m,
iVectora 'n general are herein denoted by black faced (iothic capitals, aad
|the corre.iponding unit vectors by black faced italics. For purposes of
^le.signatinn and operatioi all vectora (unless otherwise indicated! are under-
stood to start from O, th? centre of coordinates.
Then it A is any vector, u is its length, a is unit length of the some
vector, a, x, o, y, o. x are tie vector components u' A along X. Y, Z, and
«„ Oy, fi, are the lengths of these comi>onent.s.
Then A = a«
= o, X + o, y 4 ". z by vector adilition.
a= = (?; + a; -f a; I ..lid geometry.
The symbol A is used to indicate 111 the vector from to the poll ■
H'hosc rectangular coordinates are «,. n,. n, ; (2) motion from O to the
extremity of A ; (3) a rotor, defined in ST.
3S15V0
■> A CldMKTKIlAI. VMTiUl Al.llKHKA
The lini' i" I'li'U" "f A I-" ■■X|ir.'ri«vil liy nn cliinca.iil /. Ihu» ^A. iiml
liny part cif lhl» \M-m. frnm m l.i n, Id wrilli-n .,/a. Hurfufr Iwi ar.'
.irilinafily ■•t)iii««>'l li> lwi> '» uiiii »"li'l "k-I by three / ».
i. Til .Xlirfna Ih • CDsinB of thi- linitk- lii'tWfvn twu vwlorn In ti rmn (if J
• ir«inlinlilf« "f Ihi' virliiric. j
l,.t I- lif th. linitth of thi' lini' joininK the {
I'Xtri'mitici* iif the veL-lorii A. B. from t>.
Hv wiiiii Ki'omelry—
I-' -- (a, - *.!' + («, - *.l' + (o. - b,)'
♦"A
By plane IriKonometry-
f ■ -_ a' + h' - 'lab f»H A B.
Kill. 1.
n. A, + n, *, + a. h, _ S.i,
Therefore. ciM A B „*'
a b ao
where S.i. the nun of the a b produrtM - u * <o« A ■.
If S.,, — O, A B. anil conversely.
ExAMFI.K 1.— Find the angle Inaween the vectors Jt + 2y anil 'ix-y + <«.
Here S ■- 0. anil the vectors are perpeniliculur.
KxAMPl.K a.— Whatanglfsdoea the vector 2« - y +» (^ Al make with
the axes M, y. M '.'
2 -1 1
= \ •!; .'. run A « = I (j' . roil Ay ^ i t> • '"'' ^* ■■' '
4. ADDITION AND SUBTRACTION— Ailililion is ueometricall.v
ilelineil as the (inicess of makinK the second vector step from the extremity
of the first. The sum is the new vector from O to the extremity of th.>
second vector thus added.
Algebraically addition is perfoimed by lesolvinK the vectois into their
components and adding these.
A + B ^ («, * + rt, y + <l, »l + lA. * ^ It. y + b,M)
=. (n.+A.) X + ('J,+A,l y + ('/,+*,! *■
Subtraction is addition of the negative of a vector.
Hence, both geometrically and algebraically vector terms are commut-
ative.
A B ^ B + A.
A OKUMKTKK'AI. Vft:rT<»( Al UKMKA
S. COLLINEAR VECTDRS.— Two wvXum A. B. an- in th.' ^uiih' Htu-
whvn
A. b, A.
If n iH piir«itivt>, A unil ■ un- in thi' Hum*- ilircctioh . iT nvy hv*>. A and ■
rmM <if -"'*' opl""^'^''- If " '• ^ '-
IK th»- 6. COHLANAR VKCl ORS.— Thtvvv.ctont, A. B. C. iin-in thi* sunn-
)_ plum- whi-n unothvr vi-ctor, K. run Ih- fournl which ih iHTpL-mlii-uIur to nu-h
iif thi'Mi. Then S,,. - S,,K - .■',» - 0, Uy i.\. KliminatinK k,, k -v, wt-
, J lf*"t thf tuplanar i-quatiftn
* a. b, r. - I),
The dL-Wrminant ; a. A, c, in nix tim.'d the vnlumiMif ihi- tftniht-dnm wh.iM.-
comerti are O A B C. When this volunu- itt zt-io A, e ,. jirv inplunar.
E\AMri.K. Find the cnnilititinn under which A i» |n'i|K'ndifuliir to
C- « + «, and in the B C plane wh*..-e B - liJi - yi ;l.
I
i The condition of pt>r|K'ndicutarity in
S.. - W. or a, -t- tf, - 0.
y -f rM. The copianur etiualion i«
I () 1 U- 0.
O, <Iy "■
Therefore A -=^ fl. <* - yi :*- «)■
7. MULTIPLICATION of the vector B by the vector A is written
AB. and i.-* defined jfeomftrietiily as thf comliined operations,
(1) Extension of B until itH It-nKth is ah,
(*J> Simultaneoud rotation of B thru 'JD almut A as an axis, in a
direction which is riKht handed or cloekwise when facing in
the positive directi4)n of A.
V.M-h vector multipli:.'r is a tensor-rotor. The rotor power of all vectors i;*
the same and needs no separu^e expression at this »Vd^v.
The product AB is that vector from O whose extremity is tlie final
position of the [»oint B after exten.sion and rotation. The locus of AB
iA the curve traced by the i«)int B during the operation. ^ .
ABC means the operation of A on the product BC. or
Also
ABC
A'B
A. BC.
A. AB. etc.
4 A GEOMETRICAL VECTOR ALGEBRA
It next becomes necessary to finJ the laws of algebraic multiplication
that correspond to the geometric chanj^ts 'uei« defined.
S. Multiplication by a collinear vector makes no change except in
length or sign,
XX = X
XX — X
XX — X
XX — X
Aa = A. etc.
9. Unit perpendicular vectors give the following results which are
geometrically evident.
xy — M xy M
X* — y XX =^ y
xy X xy M
XX ~ y XX y
and similarly for y and m as operators.
Here the laws of signs are the same as
•^ in common algebra, so long as the factors
' are in alphabetical circular oider;
Fig. 2
xy = M - xy,
xy -- M xy =r — xy.
But reversing the order of the factors changes the sign of the product ;
xy --- m
yx - x.
The second power of an unit perpendicular opfrator is equivalent to -1,
x^y ~ X* y
x-y ~ XX y.
The fourth power leaves the operand unchanged.
x'y x-y y.
W; the vectors are not units the product of their tensors is prefixed to
th.- vt^tor product,
ax. hy~ab. xy = abx.
10. To find the algebraic product of any two vectors. Let the product
V^ ^ be K AB. Draw KD O A. and OV
equal and ;>«rallel to D K. Then the length
01) is
OD - OK ro,v AK
• b —ob co.s AB
= S bp ^'A.
^fi^ As vectors OK = OD + DK,
or K --- S„., a + V.
f-'iu. a
A GEOMETRICAL VECTOR ALGEBRA
lication
cept in
ich arc
erators.
lame as
factors
To determine V we have the equations of perpendicularity,
S„ = o, i»K + fly P, + o. r, = o
Sb. = *. i*. + ** f , + *. ''i = o,
and from the triangle ODK,
tri -\- vi -\- vi ^ V' ^ o'b' - S".
Solving we get
P, = fljAy — Oyd,.
Hence the "Vector Normal" to A,B. is
a. fly a.
b. by A.
X y X
and its length is
D= 1 a'/r'-S^= ab sin AE
The product K is thus expressed in terms of the given vectors and
their compunents, in the equation
A8 = Sa + V.
Example. Find the product AB when
A = 3y - *, B = Z* - y.
Here S = -6, V ^ 8Jt. a ^ , 10,
.'. AB 8jc
3, 10
(3y -
«).
t to -1,
product
nd OV
; length
11. PERMUTATION OF FACTORS. It i.s geometrically and alge-
braically evident that
and that v.h - — Vi,..
Hence BA -^ 8^-* + V,.
^ S„. b V.h
which is not equal to AB.
Changing the order of the factors changes the vector product. Vectors
are not permutable.
6f7 r^.
12. OPERAND DISTRIBUTIVE. To tind the product A (B ■ C)
let B C ^ D.
so that rf, - bt ' r»
dy = b, c,
d, --■ b. -^ c
6 A GEOMETRICAL VECTOR ALGEBRA
Then A (B t C) == AD
=- S^ a + V,a
= (S„ ;,_ S„) a + v^ + V,
= (S^a + v.i) + (S„a + V.,)
=- AB * AC.
The operand is therefore distributive.
13. OPERATOR NOT DISTRIBUTIVE. To find the product
(A * B)C. let A - B = K so that
*• = Ox * *.
k, = a, + b,
k, =a, ^ A.
k' = a' + W ^ 2S.t, by S3.
and
Then
(A • B)C = KC
= S^. * + v»,
S„ t_ St.
\ a' + I)' r 2 S.b
which is not equal to AC ^ BC.
Hence the operator is not in general distributive.
(A - B) + (V„ ■ v^.)
14. FACTORS MUST NOT CHANGE ASSOCIATION.
A B C = A (Sk, 6 + V^,)
= ?^AB + AV^
s ^ s
To expand A B. C, let K = A B, so that
A B. C = K C =^ Su. * + Vfcr
I fly A, I , I o, /» J , 1 o, A, I
-f- Cn Cy C,
X y *
which IS not equal to ABC.
Hence the association of a factor must not in general be altered.
But if C — A these two products become identical, and therefore
A. BA -= AB. A.
die
A GEOHETBICAL VECTOR ALGEBRA
s.,c.
15. POWERS OF AN OPERATOR.
AB= Sa + V
A'B= A. AB = A ;rfa + V)
= SA + AV
= 2 SA — o'B by expansion and multiplication.
A'B = A (2SA - a'B)
= 2 a= So - a» (Sa + V)
= o» (Sa — V), which is geometrically evident.
A'B = a'Sa- a'iSa- a'B)
= a* B, which is also geometrically evident.
From these results it is easy to write the expansion of any value of
A"B when /i is a positive integer.
16. LAWS OF MULTIPLICATION, summary.
(1) Factors are not permutable (§11)
A B is not equal to B A.
(2) The operand is distributive (S12)
A (B ^- C) = AB ^ AC,
but the operator is not (tqlS)
(A ~ Bl C is not equal to A C ± AC.
(3) The association of a factor must not be changed (§14).
A. BC is not equal to AB. C
but A. BA'^AB.A.
(4) The fourth power of an operator is equivalent to the
fourth power of its tensor (S1.5).
(5) The common laws of signs are true for operand and
product ; not for the operator.
17. PERPENDICULAR VECTORS. When A.B.C are perpen-
dicular, S.b -^ S„ = Sh, = o, IS3), and AB - V, (§15).
Then the laws of §16 become the following:
(1) Permuting the factors, i.e., interchanging operator and
operand, changes the sign of the product.
B A = Vha = - Vi* = - A B.
(2) Both operator and operand are distributive.
(A B)C = AC • BC
A (B ■ C) '^ AB AC.
A QEOMETRICAL VECTOR ALGEBRA
(3) The association of a factor must not be changed.
A. BC is not equal to AB.C.
but A. BA ^ AB. A.
(4) The square of an operator is -1 times the square of its
tensor.
A'B = - fl^'B.
(5) The common laws of signs hold true.
If a, b, e be any three unit perpendicular vectors in the
same circular order as x. y, m; then ab ~ o, bo = a,
oa = b, and these vectors may serve as units of the
system, as well ns x, y and *.
18. DIVISION is the inverse of multiplication, so that if
AB = C
f==A'C=B.
Geometrically, division is a negative turn of 90° ahout the divisor
(identical in this respect with multiplication by the negative of the divisor)
and reduction in length to that given by the quotient of the tensors.
GENERAL FORMULA for A" E
C
where n is real.
Let O A, OB, be the vectors
A. B. and let C be in line
with their vector product, so
that A" B = a" times O C.
Let BCT be the circle of
revolution of B about a; N
its centre; N B, NC, its radii.
Draw CD NB; DE II BO.
Let CNB = = n^be
the angle of rotation of B.
Then
NC =
NB
~ a'
where p
is
the
lengt
DC =
" sin D,
a
DE =
BO
ND
N"B
= b
cos
0,
; =
DB
ON
NB
~ a
vers
a.
cot
AB
^
" vera V
s
s
vera 0.
A GEOMETRICAL VECTOR ALGEBKA
i
As vectors OC = OE + ED + DC
of its
in the
9 = a.
of the
divisor
iivisor)
sectors
in line
act, 80
3C.
•cle of
a; n
) radii.
II BO.
n - be
of B.
S v
= a oers -\- B cog ff -)-- sin 8
\ .*. A"B =- (T. OC
1 = a"' (Sa vera & -^ a B cos -\- V sin 0).
This formula, being true for all real values of n. includes products,
quotients, oowers and roots of vector operators.
Example. — Two rods, A end B, are joined at one end. A ia one foot
long, and the perpendicular distance of its free end from B is six inches.
i B is turned 60" about the axis of A, then A is turned 90" in the same
^ direction about the new axis of B. Find the new position of A.
Let the joined ends be at O. Let B = bM, and A a, x + Oyy.
Since a — 1, and a, = 4, A = i (* j 3 + y).
The result of the first rotation is represented by
C = A* B -- So vers m" + B cos 60" + V sin 60"
I = ,- (7jr + y , 3- 2« , 3).
The second rotation is
c A = S c + V,.
-.= > (9x , 3 - 3y - 2«).
which gives the final position of the free end of A.
20. QUATERNIONS.
When A B, S =
and V = AB. (isl?).
Then A"B = a" (B cos « + a B sin ^)
j = a" {cos 1^ + a sin B) B.
Now a- as a perpendicular operator is equivalent to -1 ; and by ex-
pansion in series, exactly as with the complex {cos H -f / sin W), it may
be shown that the rotor of A"
cos # -H a sin 6 = c"^
where is the angle and a tht; uxis of rotation.
Hence for perpendicular vectors
A" B =^ a" e'
lO,
10
A GEOMKTKICAL VKCTUK ALGEBRA
The optrator A" is a tt-nsor-rotor-vector, or a directed quaternion,
when applied to vectors pi-rpjndicular to A. It has the four funda-
mL'ntal charicturs of a qu.iternlon, namely,
(1) Since A" B = C, A" may be regarded as the ratio of
c to B;
Ci) It is the product of a tensor and a directed rotor, a", e* ;
(3) It ia the sum of a scalar or number and a directed unlocated
line or vector, a" cos f) -{- a" a sin ft;
(4) It is a quiidnnomiiil of the form * + /* + my + nx, where
A: is a pure number and the directive units x, y, x, have the
relations
x^ = y^ -^ M- = xy* -^ — 1.
21. VECTDR ARCS. The rotor c** turns thru the angle a about
the axis A any vector in the plane perpendicular to A. The index a is
a vector angle whose axis ia A and whose magnitude is a radians. The
length of the subtended arc is aa. If this circular arc be taken as a
vector, written a, it is understood that its angle is a, its axis A and its
radius a. A vector arc may take any position in its own circle, and has
therefore ce more degree of freedom than its vector axis.
Vector arcs need not be confined to arcs of circles, but whether the
extension to other curves would be of any particular value remains to be
seen. A rou;?h classification givei H.-c follov/ing:
(1) Straight vectors,
(2) Plane vectors, having single cuivature.
A. Conic,
a. Circular, b. Elliptic, c. Parabolic, <i. Hypeibolic,
B. Spiral, etc.
(3) Solid vectors, with double curvature.
22. SUM OF CIRCULAR AND STRAIGHT VECTORS.
Let the plane of the arc a
meet the plane of A. B. in th :
line CC , let C be so chosen
that BC is not greater than
i. e., sothat n is positive ;
and let a = c.
A GKOMb:TKICAL VWTOK AUiKUUA
Let
C - mA fnS.
Thin
from
thf ti>;ure
n= b - m- a' ^^ c' = o',
ma S.,
'■'"> AS'" „b - ab
n' S
.'. n — , m —
V • 1}
.-.C-- ' (SA + a'B)
l*t
cor ' <•:
. Thtn
a^ r-D = c"'+°<
ttil
C05 (a' 4- a) -|- a S/n la' -\- a) ~ cos a'
■')0
a sin <
2 sin
(
. I ,
■Jfl/n (a'+ ^ ) + afos(fl' + ^), C.
To this B is readily added.
If B is parallel to A, C is indeterminate and any radius of the a
circle may be taken as C. In this case the sum is a point on a right helix
or screw whose axis is A. Since the addition may beKin at any point of
the a circle, the sum is a s^rew vector whose r- 'ius. pitch and direction
' are fixed.
23. SUM OF TWO CIRCULAR VECTORS. Let a. B he two cir-
jcular vectors with a common centre 0; and let C ^ V„i, be the intersec-
I tion of their pianos. Let C Bo = (i' , C Ao = «'.
A» - (e"
- Bo ^
e ) a c.
.^'+^
Any third circular vector
whose position is deter-
mined with reference to
the intersection of its plane
t with the plane of a or 6,
i may be similarly expressed
l and the sum readily found.
In expanding these ex-
pressions it is convenient to
remember that
12
A Gc:OMETRICAL VECTOR ALGEBRA
whtn C ~ V,,,
then AC = V„ = S,h A - a* B
and BC ^ W^ = tr A - S^ B.
When « _^ o the sum ia
2a ~ a. 2a
which IS a vector arc with angle a and radius 2a.
The locus of the sum of two equal vector arcs beffinning at the same
paint of intersection, when the planes are not identical, in an ellipse.
Also / la - a) is a straight tine.
24. SPHERICAL TRIANGLE. Assume a sphere of unit radiu.«,
and upon it arcs of Rrent circles. As an illustration of vector treatmtnt
let it be required to find the relation between the sines of the angles of
a spherical triangle.
Let a. B, y. be three cir-
cular vectors forming a
spherical triangle ; a, b o,
their vector axes; A', B'.
C'. the vectors from O to
the angular points; A, B, C,
the angles of the spherical
triangle.
Fig. 7
Draw A'n ' OC
Then as vectors OA' ^ On -f- n A'
or A' == C' -I- « = C cos fi 4- ^H,' sin (I
Similarly A' = B' — y — B' cos y — y,,,' sin y.
By inspection of the figure it is evident that in any spherical triangle
co.'i -- Si.i.i. ftin a = Cij.'
cos A = — cos (tt — A) ^ - Shr
, sin A = Vbc
= a\x a\ y a\ «.
A GEOMETRICAL VECTOR ALGEBRA
13
Similar equationa may be writtl'n for thi' corri'njxjndinK eli'mcntu iif the
trianRle.
From the last equation, equating coefficients of *. y, »,
I A, r, 1 1 _ i *■ ''■ I
Similarly,
. *;
s,., s,. - s,
xiny 1 1 — S:i|,i
1 c. 1. I
al =-
a. h.
Then con y — S.11.1 —
and
1 n ^^.<.^_SL -^s-:. + 2 s.i, s,„ s,.i
«/n C =' ».,, ~ 0.V Oi- 'V.
The last expresaio- is symmetrical in a. b, r. and therefore
a(n a _ "I" li mny
uln \ ~ ain B aln C '
25. CONIC VECTORS are expressible in tei-ms of the radius vector
from the focus to each e.tremity of the segment of the curve.
Let A be the axis of u con'c, O its focus,
N its directrix. P the radius vector, a. h
the coordinates of P with reference to A
and B; and let /J — e (a f m). where
1
is the eccentricity.
Fio. »
Then a cp - m
b' p= — 0-
••. P - A + B
— (cp - ml a + *, [p-'(l — c=l + 2cmp m-].
The conic vector from P„ to P is P Po .
P may also be expressed in terms of a, h. or H.
Thus a P cos II,
b — p sin H,
p-'d — (") 4- 2cmp ni-.
cp — a -\- m
= p cos H + m.
p =
m
- cos It
(a cos H + b sin 0).
14 A CEnMKTRICAL VI-TTOR ALGEBRA
If p \* a rnnntant, r - m - », thfn ftir the circle
P - /> m rnn if ■} hsin H).
Sinnliirly in any conic
A+ (.1 I (a + m>' - u^r- ;.
'■» t"'' ±_*ll*lr..]'I
a -1- B.
but when r 1, in the parabola,
p o + B,
U /«
Wf have tht-n an fxprcssiim for any cimic vet-tor iw the differenre of
two straight vertorw, P ^ Po ; which may be expressed in terms of either
i»f the viirialtles, <i. b or H.
The sum of two or more eoniL' ve'-tors would express approxi-
mately for u short distance the course of a btwiy niovinR under (fravita-
tionnl forces from two or more sources. Whether this methml of
calculatinjf would be an improvement on preoent methods 1 am not
prepared to say.
Multiplication of a straiKht ector by a vector are involves doulile
curvature, ani the locus of such a product is a convenient form by which
to »'Xpr-ess solid vectors {iiZ'\ Again the utility is problem.itical.
26. DIFFKRENTIATION t)F STRAIGHT VECTORS. Any vector,
A, may vary in lenRth antl in dir-ection. Its variation may be expressed
in terms of a for length and a for direction; or it may be expressed in
terms of the components A.. A,, A..
Since the intiiiitesirnHl inci-enunts of a vector' ai-e also vectors,
evident that by vector addition
rfA -^ ^', A + o> A + rf. A
since *. y, x, are absolute constants.
it is
A Is.:
(/A
, A 4 r'.A
da \ a f/f/.
(2)
It follows from (1) that tht: differential of a vector i^ the sum of the
differentials of its components, and hence that differentiation is distrihu-
live over vector terms.
A nEOMKTKICAI. VKCTOli Al. iKHK*.
IR
It follow-* frn-n |2) that thi' oT-ilinary rulf for dilTiTt'ntitilion vf u
pro tui-t holili* truf for any unit v*'i-l(n- uml itn tt-nnor, anil hent-f ftjt any
priHluct uf :t tt-nHor and a vector.
27. To timl till' (litTL-rfntiiil cofffli-ii'nt of a vector procliict. A"B.
A" B 0" ' (Sa rers // -f « B ■om « f- V j»//» h).
I)iiri.'r-i>ntiatjnt( both sides uf the etjualion with reHpi>t-t to tf, n ' ,
2
ti
A"B loyn + «" 'iSa
.-(/« If -\- V r«N /y)
A"B Mtfrt (- «" ' {CO.-. H -\- a sin H) y,
:4ine(>, hy multiplicatxjn, Sa u B - a V-
ThL' l;Ht term of the (lilTfrential t-oerticient may also he written
«" ' e^** V. It is a tenwor ami rotor prixluct of V (the veetor normal
I'f A and Bl. whose lotatUm is about A.
V expresses tht rotation of
V in the plant' perpemlicular to A. a' ' o" V traces a spiral in this
plane, and as a vector it ifives at any |K*int the directiitn and rate of
motion in this plane made by the point B. supplementary to the increase
in '.he length of B.
2
The term _ A"B/o»n is a multiple of thi- vector product, and for
any niven value of n it expresses the rate and direction of the increase,
in lenRth only, of that produtt.
The aum of the two terms ^ives the rate and direction of the motion
of the point 8 for unit increase in H. It is the vector tanKent to the
curve traced hy A"B. namely, the curve ^A"B.
KXAMPLE.— Find the tangent where the flat spiral {axy\ by cuts
the Y axis. The tangent is
, (rtJrr by.
2 2
— a" b \(Z logo cos II - sin>t)y + i ^ lug a. .sin ft -
At the starting point // ^ o, and
2
Tn -^ b {__ Iny ay t *)■
When n -=2, H ^ v. and
T, = - fl- T„ .
'H)m]
in
A (iitniirTiitr«t vrrroii AinRimA
Whfn n 4. II 2ir. anil
T, <i'T, - .I'T,.
Thl« vi'ftiir liinif'nl muki'a Rt nil tinifn a conntant anitir with it« rmliuii,
and ill li-nitlh Kiv.-ii tho vslorlty of thu generatinn pol"' when the
aniniliir vt'ljvity it unity.
M. CURVATURE. If / be the length iif a curve, anil T the vector
tanuenl, the curvature !< la ., , anil the railiu!* of clrvature ia
29. LINEAR LOCI are loci having only one degree of freedom;
iinea or dihcrete points. A few examplea are given:
a..l^ B is anv part of the straight line drawn from the
point B in nhe direction A.
, „l A"B + C. vhen A B. is a circle with centre C. radius
b, and plant- perpendicular to a.
/
' A CON H + B sin m
A B plane.
C in an ellipse parallel to the
[<A"B + / C) includea a variety uf curves.
If A B. C II A. and a — \, the Iocuh is a helix.
If C ' A the locus varies from a circle (whtn r — ol
to a straiRht line (when c = xi, passing thru the cycloid.
In other p-nitions of C the helix is acute anRled. When
a 1 the curves are expanding and when a 1 diminishing.
30. EXAMPLES OF SURFACE LOCI.
"'"•''^ Bl ■*- C is a paialk-lujcrjim whose adjacent sides
A. B. start at the point C. Its diagonals are A • B.
Ita area is ab sin A B — fj. (-ilOi.
A OKOMKTHIiAl. VKCTOK AUJfBKA 17
„/,,/*" a* • i« a cIoMwl lurfaci', Mphi-ruul if a *. with
riMliuH b.
.,1 „la" B tM thu conical surface trnct-d l>y B hk it ix turnt'il
ttliout A.
./:/'.
(A + < 1 a) Bl i> tht' trianicU' <> A H.
31. EXAMI'l.ES OK SOLID LOCI.
Ill I A + B -I- C) f D iii any imnillelopiiiMl.
Its dl»itoniil« ari' A + B + C. A + B C. A B C.
- A 4- B + C. Il« vcilumi' in I a, ft, r, I . If P V„
■nd Q v„, thi- dihmlral anitli", «. ovir thf tditi' A ii
found from the liquation S^~PQ cos a .
J „l , I "' I'" A i» a "'"■'I' "phi'ricul if A B.
^l a" , „l*"\ / C + R I i« a hoMow annulus
B C. R C. A R. A B.
32. THE fiGION COMMON to two looi in found liy equntinK the
coefficientH of x, y. M, in the expreHsions for the loci. If thet*" e<iuutionD
are consistent. KivinK real values for the variiihles. the limits thus found
are inserted in either of the loci to irive the required locus of intersection.
ExAMPl.K 1.— Find the region common to the straij^ht line
^Inx -\- iy, and the curve
„/ |a * f (2« + yl tin a \.
Equating coefficients,
n = o -i- 2 .t/n a
i — fiin a.
Whence n = a + 1 — arcsin i + 1
Inserting these values, both loci become
,tli V + * orcsin il
which is a row of discrete points parallel to x.
'' A GEOMLTHICAL VKCTOR ALGEBRA
Example i -Fiml what part of the helix
„/(«"y + 3n X) -- ,7(3/1* +y cos -~+ a „„ "^)
is within tht- figure
JJLy-cM+bx - y) - - J'lfl'icsln'"^ +b,-y+,ccos
Efjuating coefficienta of x, y, jr.
;i 1
(1) 3/1 - c s(/i
(^) cos ~ - -.
- 1.
. sin
= 0, and /I =2, 6, 10
111 /wr nir
|J» c cos — = s/n
■i 2
If c = 0, 3/1 = •
COS-;- = 0, s/nl-^
2 2
I, and .-iincc c is positive 3n 6 -f- c.
Inserting these values in the locus of the helix we get for the
intersection a row of points
r.inx — y, where n has the values
2,6, !0 up to--t^.
Example 3.-Fiml the intersection of the plane
II {mx + nx) + 3x
with the solid
J J.lx (ax + by) =^-,rjJ[„x + hycosu hx«ino\.
Equating coefficients of x, y, x,
(1) /n + 3 = a. or /n — fl - 3,
(2l b cos H - 0, .-. b 0, or sin H I.
(3) n = h sin II, — or h.
.: n = _. b.
Suhstituting in the locus of the plane we get for the inter-
section the parallelogram
ijl"* !'I)
A GEOMETRICAL VKCTOR ALGEBRA 19
EXAMlM.E 4.— Find thu locui of the inUMsec-tion of the cube
,.l.,l.,l ln.x+ a, y a, z)
with a plane which cuts its diuj^onal
^ ~ * + y + * perpendicularly.
Let B — X - y
he one vector in the jierpendicular plane, and
C = V.b == X + y - 2*
the other. The plane in
//(/ B /hC+ nAI
where n is an arbitrary constant expressing the fractional
distance from O to the point where the diagonal is cut.
Equate coefficients of x. y, z. in the two loci,
«, ^^ / m + n
a, = -/ + m + n
a,~ — 2 m n.
Therefore
2
Ox -f ffy
3n =
G
fly + ff,.
If n _ the plane goes thru O. Since a,.a,. a„ are all positive
ami the sum zero, each of them is zero, and the locus of
intersection is the point O.
If n = 1 the point of intersection is A.
If n = h. so that a, a, a, ^- 1, while each vaiies between
and 1 subject to this condition, the locus is an equilateral
triangle whose cornos are found by giving to o„ a., a,.
separately the maximum value, 1, in the expanded .xi.res-
sion for the plane
m\
If n = S the locus is a similar tliangle.
If n = J the locus a regular hexagon.
C I
20
A GEOMETRICAL VECTOR ALGEBRA
33. PR(MECTIONS. To expres, any vector K m tcrn.s of three „.,„.
coplanar vectors A. B. C, write
/A m B ■ n C = K
•■• / o, . m A, nc, = *_
la, m b, nc, ^ *,
'a. mi. nc, = *.
. _ j_*. A, r^
I o. A, c, I
I *, a, A. [
I ". *, f, I
If we now write n 0, /A + m B is the projection of K, made
parallel to C. upon the plane of A, B.
If A and B only are given, and the projection is desired of K per-
pendicularly upon A. B. take C = V.h = , <,. A, z I . and proceed as before.
To project K in the direction of C upon a plane pe.pendicular to C
take any vector A. . c. so that S„ = 0,
as, A = c,x - r, y
and a second vector B, - Vh.. .
[ ^1 ''y C, I
B = } e, -c, I .
\ * y *\
Then express K in terms of A. B. C. as before.
The most general form of a locus is /// K + m. which is projected
in the same way.
Example. -Project upon the YZ plane and parallel to D the helix
* S ~ ./(""by + anjc)
„l \l> (y cos H + I sin «) + o n ;, | .
Let B = /y -f- mx rO.
Equating the coefficients of x, y, »,
r <l, — an
I -i- rrfj, = A cos H
ni + rd, = b sin H.
■■■ B' = y lb cos W - fl« ^'' ! + , |A .„>, H - „„ f I
the locus of which
is the required
projection.
A GEOMETRICAL VECTOR ALGEBRA _1
34. PLANE ALGEBRA. Every vector in the X Y plane is uf the
'<"'"' A = o, X + o, y
= n. X + o, »«
= (0. + X a,)*.
Since X is a part of every vector expression of this form, it may be
om.tt.-d. The remaininK form, a. + x o„ is a complex number. Sine x»
a, a rotor is equivalent to -1. «re may write this tensor-rotor in the
common form,, + ,b (wh.-re (^•=-1). who.se properties are well known.
A;?ain. any vector in the X Y plane may be expressed as a x-proiluct
thus, A = ax"x
= a ico.s H + * sin H) X
= a c'» X.
Omitting X as before we have left ,e other two forms of the com-
plex number.
Vector multiplication in the XY plane with any other rotor than z
Rives in general imaginary products, i.e., products lying outside of that
plane.
FOUR-SPACE ALGEBRA
35. In four-space there are, by definition, four mutually perpen-
dicular axes, X, Y, Z, U. These are .,o selected that they multiplv in cir-
cular order, as in 3-space. tach vector is now fully defined by four
cj.Tipjn^nts. Vectors are added and subtracted as in 3-space.
As in S3 it may be shown that
S.I, E=; o.A, -|-o,A, -h o.A, -|-o„*„ = aAcos AB,
where S.,, is, as before, the sum of the a A products.
Evidently also when S,,, - 0, A B.
36. MULTIPLICATION in 4-space is defined as rotation about the
plane' of the multiplying ve.'ors, thru a right angle in the positive direc
t^on^JThe planes of rotation are whollyt perpendicular to the axial plane.
'Rotation is essentially plane motion. In a 2-flat the axis of rotation
IS a pomt. In a 3-flat the axis i.s a line. In a 4-flat the axis is a plane
"i"' "1 4-space absolutely perpendicular planes
R /, 1"",. r "- ' + "'-^ '•" "ny ^tor in the X Y plane, and
B=~ A.x -(-*.u 18 any vector in the ZU plane. Since S ,= (l A ■ B
Tectlr'in'th '7^,', T"^ "■■"'"' '" ""■ ^^ "'"'"= '^ perpendicular to every
»'ector in the Z U plane. '
+It is evident from (J35,
exist.
^* * GEOMETRICAL VECTOR ALGEBRA
Multiplication of C by AB, ia wntten J^hc. and is defined as
(1) Rot.ti.,n of C thru 90" i„ the positive direction about the
plane A B, and
V2) Simultaneous extension to the lenjith abc.
By delmition, iTyx = «, r;„ _ , =7; . —
* *•■ — *, MUX = y, ujty ^= X
plane rZT"' ''!:' "" "'™' "' '■'"^''™ '^ -"P-^-'- ,„ the axial
Plane it becomes evident that
Coplanar vectors are unchaoged in position by 4-space multiplication
because the whole axial plane is unmoved, "P"eation.
*J'l<'x* +a,y) =a.x+a,y.
37. MULTIPLICATION BV PERPENblCULAR VECTORS.
Let A B C, and let AB C = W.
Then S„ = S^. = S,., =
and l^= IV} + 0,^ + ^1 i u>;=^ a'ly' c'.
Solving for «,., „,„ „,„ „^, a„j collecting,
tW(
Aiso
W.k.
[ o, (Jy a, a^
. j A. A, A. A.
I C^ C, C. Cu
I ' y < u
■ I o« A^ e, u I
»' = I a, A. c„ I + I a, A_ P^ I , ^ „_ ^_
s~ s.^ s„ :
S.. Sn Sk, : =- o' A' r'.
S„ S.k S„
f . I ■ + I o. A,
A GEOMETRICAL VECTOR ALGEBRA
23
>out the
le axial
38. C is coplanar with A. B, when
C — m A ■ nB.
Wntins tho four equations of coordinates ami „Umi .■
Ket the coplanar equations "i'm.nat.ng n, and ,, we
I o. b, c, , =
I o. A, c„ , = 0.
39. To find the perpendicuiar N., fr„„ th, p„,.„, g ,„ ,^^ ^^^.^_^^ ^
"^^ 1 B Let Q B be the positive direction of N,.
Then </ = OQ = A roa AB = -■"
a •
S ,
■ • a = 7, A.
A B I + ° •
i S.. S.k i , ..
|S„. S,„' *'''=o^-
These forms of N- n^ es -j
- ..ens. .. Jwd^i:;:::'::;— ::.-• -- =-
I *. *,
40. To find the perpendicular N, from c to the AB plane.
Let Q C be the positive direction of N-,.
Draw Q E ■ A O f.^ o ■ • „
^ "• "^ ' B. join C F, C E
Then OE =
a a
■■■ S.„=.s..
and O F == §ia ._ S,.,
* A •
■■• S„„-. s,,
Since A, B, Q,, are coplanar
I a. *, I?, ' =
(1)
(2)
. (3)
4).
24
A GEOMETRICAL VECTOR ALGEBRA
Solvinit for the coonlinatfs of Q,, and collecting ti-rma.
where
Then
I
ii ' »; 17; «;
la'S,.,
S.,,S.,.i
= (o'SL ■ A=S:, - 2S.,,S,„ S|,,l
o3 = o»*' -Si, .
N, = C-
S.. S.I, s.,
s,,. s,,,, s,..
ABC
S.. S., I
Si. S„i, i
I s„ S.,, S„ I
n3 = c" - »; = I S„. Sh, S„ i
I S,. S,k s„
I R«, S.I, I
I Sk. s,„, I
These forms are identical for i!-space, and apparently for all space
above it.
In 3-space also
_ I g, *, r, [
41. To find the normal N.. fiom D 10 the 3-tlat of A. B.C.
Let QD be the po.«itive direction of N,- Ji in
OQ. Drop perpendiculars from Q. on A.B. C.
and join each point of intelsecticn with D.
Then it ia evident as in
NU that
S„=S., (II
Sb, = Sm (21
S„= S„, (31.
Since A. B. C. Q, are
all in the same3-flat and
therefore all pei|Kn-
dicular to N4
S..=S„.= S,„-. S,„=(l.
Eliminating the n's we
get the cosolid eqiat'on
I ".*,<■.?„ i -0 , (4l.
A OEUUETRICAL VECTOR ALGEBRA
SolvinK for q, etc. and collecting terms
Q. =-!s.tAiS£. -A-a +B(c--s..-s„.s., . c(A's„ - s.s.)]
+ S.,rB(Sii,-,»c-)+C(»-S,-S.,S.I+ Air's,, _S^S )]
+ S.4C(SJ„-a"*», + A,4'S.-S,S..)+B(a'S.-s...sI.]|-H,^
Therefore _ _
■u - r, « S,. St. St, sL\ \^~ S" S.,
N.= D-Q,= ^ „ „ " + u o o
Ci « r. « I ^ I »!« &bl. Obc
S,,. S,t S„
s. S^ s„ s.,
s,. St. St, St,
S™ S,t s„ s„
A B C D I
/« =
I S „ Stt S,, S,.|
I S„ St, S.,, I
I a. *, <•, rf„J
W.
V.
42. RELATION OF N, TO THE RECTOR W. Since N. and W
" '•"''''""™""" '° ''" '■'""°^^- «■=• 'hey differ only in their
Hence N. = - W= '"■*>'•.< I
Similarly, in 3-space
And in 2-space
^1 _ "'- \o, b,
Nz - ^ F - ^,
when F ".a.
The form, for N.N,. „f, „?, may be obtained by suppressing rows
and colunms m the determinant fo.ms of N.. n',. It is evident that we
have here a correspondence between the geometric space-form for a per
pend,eular and the algebraic space-form or matri., which is true f„r a ,
F,
' is the perpendicular to A.
4... To find the product Ab"c when n is real.
/,
.>*="'
Let CPK be the circle of rotation
of the point C. and let Q be its centre
in the AB plane.
Join QO, QC, QP.
Let O P be the position of DC after
rotation, so that OP=JB"c.
Draw PD i QC. DMIICO.
The angle CQP -= <; _ a""
2 ■
3S
A OEOMETRICAL VICTOR ALOEBRA
Then ,i„ec. PD, being in the plan, of rotation, i. perpendicular to
the A B plane, and al»o to QC; PD is perpendicular to the S-fl.t of
A. B. C, and is therefore parallel to W.
MD- OC^^ = ceo»tf
DP = Pd sin U^n,«l„e^1'-,/„H.
An vectors
0P=- OM . MD+ DP.
■ •■ She - Oi Ders H +C cos e +
W
sin fi
and
44. If C is perpendicular to A and B. then
AB°C = o" *" j C costf + ^ «,„<,;,
.-. AB C = o"A" (C9S H + ab sin i9) C
= a'b-e^O C.
The rotor «=»« resembles the rotor e"" found in 3-space multiplica-
tlon. It IS evident that similar rotors (quaternions) will be found in all
higher space forms.
45. The intersections of loci are found as in S32.
Example 1. Find the intersection of the 3-flat
/// (ox + Ay + cjt)
with the holix
Il^"''-+y + *) + nX<~l,j, . y + ,,„,y + „ ^,„ „ _ „^j^
Equating coefficients of x, y, », u,
a - n ^ 1
A - 1
c = cos H
u = sin 0.
■: c = ^ I
and n - 0. + 2, i-, 4, etc.
A GEOMETRICAL VECTOR ALGEBRA gf
The interaection ii
i\n+l)M+y , M,
representing two row, of poinU parallel to X.
'I «ll ir.» • c,y . (r, COS # - c„ sin mm
Tfl ■ <<^uCosH • c.afnmul
Equating coefficients of x, y, ''
o — <; + m
and the plane locus becomes the rectangle
J.. I lie. + mtx + c,yl-
vectors ^sToT- "" ^""'^ ""^ ^^"" ^ ■" '"™' »' »^ f°-
vectors A, B, C. D, not in one 3-flat, write
/A+mB + nC • rO - K.
Then la, + ml>. , nc, + rd. = k,
la, + mb, + nc, + rd, r= *,
'o. + mb, ' nc, + rd, = *.
Ia„+mb^ ■ nc^-^rd„ ~ k^
1 =
i Oi *, c, rf, I
J^o, A, *. ft, I
la, *, c. rf, j^
1 "< *, c, rfTT
Wnt,„g ether /, n,. n o. r equal to «ro the remaining terms „f K
are he projection „f K made parallel to the vanishing vector and upon
the 3.flat of the remaining vectors. To project K normally upon the 3-flat
of A, B, C. write D =. W..., then make r = 0.
The sum of any two term, nf K is th. projection of K u|>on their
plane, made parallel to the plane of the other two vectors'.
Loci are projected in the same way.
28 A OEOMKTWrAI. VrCTOR AI.Or.BRA
47. A» an lllu»tration of the mi'thwl of the lut wriion we may And
the prmcipal orthogonal projections* of the regular 8-cuhed tewaraet
whole edge i» unity,
./;/:,/:/k.
upon the S-flala almut it.
(1) Parallel to ,, on the S-flat of y. ,. u. the projection i.
obtained by writing *. - 0, giving the cube
.l.l.hii,y + k.M + k,u).
(2) Parallel to * y. Let « + y — D.
To get three other rectangular vectors we may take
A = «
B = u
C - W.bd = y - «.
Then / = *., m = *., n = ^~-*' .
Writing r = the projection becomes
,/:/;/,:/'i*.A+*.B.*-^-ci.
And a = 1, 4=1, e = i 2.
To express this locus in geometrical terms we note first that since
it contains three vectors, not coplanar, with independent
variable coefficients, it is a 3-space solid ; and in the second
place that the original axes. «, u, which are perpendicular
to the line of projection, remain unchanged. The axes x. y, are
each foreshortened in the ratio of , 2 : 1. Projecting * and
y by the same plan as for K we get for the projections
.*' - J C
y' = J c
mai<ing the total distance , 2 along C.
Consider next the variables in the locus. *. and *. are entirely
independent, with limits from to 1, and ,/'/'(«■. A • * a)
is a square in the A B plane. The solid " is' a right square
prism whose extension along C is given by the last term
-C of the ln<-u .«. *, ami *■. vary independinlly from
K«ZcJZ!»ll ^^"Z'"" '■"™''iK-«™ of these projections see the
AMERICAN Journal of Mathematics, Volume XV, No. 2, pages 179-189.
A OEOMRTIIICAL VICTOR ALOEBRA ■
to I. The lower limit of the term oecun when *, - 0. * ^ I
n.m.ly, -JC: .nd the upper limit I. JC. Since'c^', j. ,hj
length of the priim la , 2 along C.
(3) Parallel to * + y i «, (_ o).
Take for the other recUngular axea
A = «i
■ -«-y
C = Wuj = * : y — 2a.
Then /=*..„=*■ -.*r_ „ ^ *!L+_5._i*.
Put r= 0; t!ie projection la
where = 1, A = , 2, c = , 6.
The figure i. again a 8-.olid : thu axia „, perpendicular to D. remain-
ing unchanged. Projecting the other three axea we get
«' = iB + }c
*■' = - J B + } C
«' = - J C.
The length of each of theae ia ^. This length may be found directly
by the equation
s;,,
ain' jiO= I - cos'xa 1
The variable t. is independent. The figure i. therefore a right pri.,m
of unu ,ngth along A. To find the pri™ base, or section in
the BC plane, draw the axea i B, il C, and plot the figure.
First, let *.=*, = 0, while *■.
varies from to 1, tracing the
line along C from - j to O, the
line oO. Next let *, 1: the
locus of *. is then the line be from
B I B , C
2 6 "-2+6 ■
Intermediate values of *, fill out
the parallelogram ac.
A OKOMKTmcAL VECTOR AU)»RA
Ntfxt li<t *, - 1, BndiiroNud u b«fur», obtaininx the parallrloKnni rfC,
whow llmltinR liriM ore rf« from" ^ to " f , «ncl OC
from to y ,
lnti.™*.liate v-lUM of *. Klve .Imll.r parallelogram- commencinit at
evi-ry point alottR a,l and coverinn the regular hexagon nr,.
The whole projection i« a right hexagonal prIam. The projected axe«
*', y', M\ are Or, Or, Oa.
(4i Parallel to m + y + m + u, (- O)-
Take for the other rectangular axea
A My
C = i W.U = « + y - « _ u.
Then /=*■;*',„ *•:*",„ " ■ + *,- *■ -^ *■
and the Incus of the projection ia
mil
2 A + 2 ■ + -T-' (
where a — l> — , 2, c — 2.
Projecting the axea «, y. ». u. we get
*' i A + J C
y'= JA + ic
«'=JB IC
u'^ - JB JC
and the length of each projecttd axia is
To obtain the geometric form of the projection, give t<, all the variables
the value zero, then to each one separately giv.. all values up to
unity. This gives four lines from O, identical with the projected
axes. With three of these lines as adjacent edges form a
pirallelopiped. and for.n three more parallelopipeds with the three
other possible groups of the four lines. The sum of these four
solids, a rhombic dodekahedron, is the projection required.
3